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IEEE Conference on Decision and Control (CDC)

Two-stage solution of quantum process tomography in the natural basis

Shuixin Xiao, Yuanlong Wang, Daoyi Dong, Jun Zhang

Published Dec 6, 2022• 3 citations
Publisher

Abstract

Quantum process tomography (QPT) is a critical task for characterizing the dynamics of quantum systems and achieving precise quantum control. In this paper, we propose an effective two-stage solution (TSS) for QPT in the natural basis. Our algorithm has O(d6) computational complexity where d is the dimension of the quantum system. We establish an error bound $O\left( {\frac{{{d^5}}}{{\sqrt N }}} \right)$ where N is the copy number for each output state. The numerical examples compared with a convex optimization algorithm demonstrate the effectiveness of our algorithm.

DOI
10.1109/cdc51059.2022.9992920

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Quantum process tomography (QPT) is a critical task for characterizing the dynamics of quantum systems and achieving precise quantum control. In this paper, we propose an effective two-stage solution (TSS) for QPT in the natural basis. Our algorithm has O(d6) computational complexity where d is the dimension of the quantum system. We establish an error bound O(d5N)O\left( {\frac{{{d^5}}}{{\sqrt N }}} \right)O(N​d5​) where N is the copy number for each output state. The numerical examples compared with a convex optimization algorithm demonstrate the effectiveness of our algorithm.
Quantum process tomography is a critical task for characterizing the dynamics of quantum systems and achieving precise quantum control. In this paper, we propose a two-stage solution for both trace-preserving and non-trace-preserving quantum process tomography. Utilizing a tensor structure, our algorithm exhibits a computational complexity ofO(MLd2)O(MLd^{2})O(MLd2)where d is the dimension of the quantum system andM,L (M≥d2,L≥d2)M, L~(M\geq d^{2}, L\geq d^{2})M,L (M≥d2,L≥d2)represent the numbers of different input states and measurement operators, respectively. We establish an analytical error upper bound and then design the optimal input states and the optimal measurement operators, which are both based on minimizing the error upper bound and maximizing the robustness characterized by the condition number. Numerical examples and testing on IBM quantum devices are presented to demonstrate the performance and efficiency of our algorithm.
Quantum process tomography (QPT) is a fundamental task to characterize the dynamics of quantum systems. In contrast to standard QPT, ancilla-assisted process tomography (AAPT) framework introduces an extra ancilla system such that a single input state is needed. In this paper, we extend the two-stage solution, a method originally designed for standard QPT, to perform AAPT. Our algorithm hasO(MdA2dB2)O(Md_{A}^{2}d_{B}^{2})O(MdA2​dB2​)computational complexity whereMMMis the type number of the measurement operators,dAd_{A}dA​is the dimension of the quantum system of interest, anddBd_{B}dB​is the dimension of the ancilla system. Then we establish an error upper bound and further discuss the optimal design on the input state in AAPT. A numerical example on a phase damping process demonstrates the effectiveness of the optimal design and illustrates the theoretical error analysis.
Quantum Hamiltonian identification (QHI) is important for characterizing the dynamics of quantum systems, calibrating quantum devices, and achieving precise quantum control. In this paper, an effective two-step optimization (TSO) QHI algorithm is developed within the framework of quantum process tomography. In the identification method, different probe states are input into quantum systems and the output states are estimated using the quantum state tomography protocol via linear regression estimation. The time-independent system Hamiltonian is reconstructed based on the experimental data for the output states. The Hamiltonian identification method has computational complexity O(d6), where d is the dimension of the system Hamiltonian. An error upper bound O( d3/√N ) is also established, where N is the resource number for the tomography of each output state, and several numerical examples demonstrate the effectiveness of the proposed TSO Hamiltonian identification method.
Quantum process tomography (QPT) methods aim at identifying a given quantum process. QPT is a major quantum information processing tool, since it especially allows one to characterize the actual behavior of quantum gates, which are the building blocks of quantum computers. The present paper focuses on the estimation of a unitary process. This class is of particular interest because quantum mechanics postulates that the evolution of any closed quantum system is described by a unitary transformation. Unitary processes have significantly fewer parameters than general quantum processes (22nqb vs. 24nqb−22nqb real independent parameters for nqb qubits). By assuming that the process is unitary we develop two methods that scale better with the size of the system. In the present paper, we stay as close as possible to the standard setup of QPT: the operator has to prepare copies of different input states. The properties those states have to satisfy in order for our method to achieve QPT are very mild. Therefore, we choose to operate with copies of 2nqb initially unknown pure input states. In order to perform QPT without knowing the input states, we perform measurements on half the copies of each state, and let the other half be transformed by the system before measuring them (each copy is only measured once). This setup has the advantage of removing the issue of systematic (i.e., same on all the copies of a state) errors entirely because it does not require the process input to take predefined values. We develop a straightforward analytical solution that first estimates the states from the averaged measurements and then finds the unitary matrix (representing the process) coherent with those estimates by using our analytical solution to an extended version of Wahba’s problem. This estimate may then be used as an initial point for a fine tuning algorithm that maximizes the likelihood of the measurements. Simulation results show the effectiveness of the proposed methods.
Quantum process tomography (QPT) methods aim at identifying a given quantum process. QPT is a major quantum information processing tool, since it allows one to characterize the actual behavior of quantum gates, which are the building blocks of quantum computers. The present paper focuses on the estimation of a unitary process. This class is of particular interest because quantum mechanics postulates that the evolution of any closed quantum system is described by a unitary transformation. The standard approach of QPT is to measure copies of a particular set of predetermined (generally pure) states after they have been modified by the process to be identified. The main problem with this setup is that preparing an input state and setting it precisely to a predetermined value is challenging and thus yields errors. These errors can be decomposed into a sum of centered errors (i.e., whose average on all the copies is zero) and systematic errors that are the same for all the copies. The latter is often the main source of error in QPT. The algorithm we introduce works for any input states that make QPT theoretically possible (i.e., unless there are several solutions due to, e.g., a lack of diversity in the input states). The fact that we do not require the input states to be precisely set to predetermined values means that we can use a trick to remove the issue of systematic errors by considering that some states' copies are unknown but measured before they go through the process to be identified. We achieve this by splitting the copies of each input state into several groups and measuring the copies of the kthk\mathrm{th}kth group after they have successively been transferred through kkk instances of the process to be identified (each copy of each input state is used for only a single measurement). Using this approach, we can compute estimates of the measured states before and after they go through the process without using the knowledge we might have about the initial states. We test our algorithm with simulated data, and we assess its performance with a cnot gate on a trapped-ions qubit quantum computer.
Quantum Process Tomography (QPT) methods aim at identifying, i.e. estimating, a quantum process. QPT is a major quantum information processing tool, since it especially allows one to experimentally characterize the actual behavior of quantum gates, that may be used as the building blocks of quantum computers. We here consider unitary (possibly dense, i.e. without sparsity constraints) processes, which corresponds to isolated systems. Moreover, we develop a QPT method applicable to a significant number of qubits and hence to a high state space dimension, which allows one to tackle more complex problems. Using the unitarity of the process allows us to propose a method that first achieves part of QPT by performing an eigenanalysis of the estimated density matrix of a process output. We then complete the QPT by using another state differently. The relevance of our method is validated by means of simulations.